# Prove a_0+a_1+…+a_2016>3^(2017)−1

• MHB
• lfdahl
In summary, the conversation discusses the properties of a polynomial with 2017 real roots and another polynomial with no real roots. From this, it is proven that the sum of the coefficients of the first polynomial is greater than a specific value. This proof is achieved by considering the minimum value of the second polynomial and the range of its roots. Overall, it is shown that the sum of the coefficients of the first polynomial is greater than 3^2017-1.
lfdahl
Gold Member
MHB
Suppose, that the polynomial $P(x) = x^{2017}+a_{2016}x^{2016}+ a_{2015}x^{2015}+ … + a_1x + a_0$ has $2017$ real roots,

while the polynomial $P(Q(x))$, where $Q(x) = \frac{1}{4}x^2+x-1$, has no real root.

Prove, that $a_0 + a_1 + … + a_{2016} > 3^{2017}-1.$

lfdahl said:
Suppose, that the polynomial $P(x) = x^{2017}+a_{2016}x^{2016}+ a_{2015}x^{2015}+ … + a_1x + a_0$ has $2017$ real roots,

while the polynomial $P(Q(x))$, where $Q(x) = \frac{1}{4}x^2+x-1$, has no real root.

Prove, that $a_0 + a_1 + … + a_{2016} > 3^{2017}-1.$
[sp]If $\lambda_1,\,\lambda_2,\ldots,\lambda_{2017}$ are the real roots of $P(x)$ then $P(x) = (x - \lambda_1)(x - \lambda_2)\cdots (x - \lambda_{2017}).$

If there is a real number $x$ such that $Q(x) = \lambda_k$ for some $k$, then $x$ would be a real root of $P(Q(x))$. Therefore none of the roots $\lambda_k$ can be in the range of the polynomial $Q(x)$.

The minimum value of $Q(x)$ is $Q(-2) = -2$, so the range of $Q(x)$ is $[-2,\infty)$. Therefore $\lambda_k < -2$ for all $k$. It follows that \begin{aligned}P(1) &= (1 - \lambda_1)(1 - \lambda_2)\cdots (1 - \lambda_{2017}) \\ &> (1 - (-2))(1 - (-2))\cdots (1 - (-2)) = 3^{2017}.\end{aligned} But $P(1) = 1+a_{2016} + a_{2015} + \ldots + a_1 + a_0$. Thus $a_0 + a_1 + \ldots + a_{2016} > 3^{2017}-1.$
[/sp]

Opalg said:
[sp]If $\lambda_1,\,\lambda_2,\ldots,\lambda_{2017}$ are the real roots of $P(x)$ then $P(x) = (x - \lambda_1)(x - \lambda_2)\cdots (x - \lambda_{2017}).$

If there is a real number $x$ such that $Q(x) = \lambda_k$ for some $k$, then $x$ would be a real root of $P(Q(x))$. Therefore none of the roots $\lambda_k$ can be in the range of the polynomial $Q(x)$.

The minimum value of $Q(x)$ is $Q(-2) = -2$, so the range of $Q(x)$ is $[-2,\infty)$. Therefore $\lambda_k < -2$ for all $k$. It follows that \begin{aligned}P(1) &= (1 - \lambda_1)(1 - \lambda_2)\cdots (1 - \lambda_{2017}) \\ &> (1 - (-2))(1 - (-2))\cdots (1 - (-2)) = 3^{2017}.\end{aligned} But $P(1) = 1+a_{2016} + a_{2015} + \ldots + a_1 + a_0$. Thus $a_0 + a_1 + \ldots + a_{2016} > 3^{2017}-1.$
[/sp]

The ink has hardly dried, before you came up with this excellent solution, Opalg!
Thankyou very much for your participation!(Clapping)

## 1. What is the purpose of proving a_0+a_1+…+a_2016>3^(2017)−1?

The purpose of proving this inequality is to show that the sum of a finite geometric series is greater than a certain value. This can have applications in various fields of science and mathematics, such as in calculating the growth rate of a population or the interest rate on a loan.

## 2. What is a geometric series?

A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant value, called the common ratio. The sum of a geometric series can be calculated using the formula: S = a(1-r^n)/(1-r), where a is the first term, r is the common ratio, and n is the number of terms in the series.

## 3. How do you prove a_0+a_1+…+a_2016>3^(2017)−1?

One way to prove this inequality is by using mathematical induction. First, we can show that the inequality holds for the first term, a_0. Then, we can assume that the inequality holds for the sum of the first n terms, and use this assumption to prove that it also holds for the sum of the first n+1 terms. This will show that the inequality holds for all terms up to a_2016, thus proving the statement.

## 4. What are some applications of proving this inequality?

Proving this inequality can have practical applications in various fields such as finance, economics, and biology. For example, it can be used to calculate the growth rate of a population or the interest rate on a loan. It can also be used in engineering and physics to analyze the behavior of systems that follow a geometric growth or decay pattern.

## 5. Can this inequality be proven using other methods?

Yes, there are other methods that can be used to prove this inequality, such as using mathematical inequalities or calculus. However, the method of mathematical induction is commonly used for proving statements involving sums of series.

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